The matrix manifold \(\mathcal {M}\)  is dictated by the known physics of the system at hand.
For example, we may select \(\mathcal {M}\)  such that its members satisfy certain symmetries that we known are obeyed by the system at hand.
We call (REF ) `physics-informed DMD' (piDMD) as the optimization integrates
underlying knowledge of the system physics into the learning framework.
Again, the spectral features of the solution of (REF ) can be computed to give insight into the modes that dominate the system and their dynamics.
The low-rank DMD regression (REF ) is a special case of piDMD where \(\mathcal {M}\)  is the manifold of rank-\(r\)  matrices.
However, piDMD models are not always low rank (see, for example, §REF ) but nevertheless have few degrees of freedom due to the physics-informed constraints.
Constraining the solution matrix improves the generalizability of the learned model,
reduces sensitivity to noise and reduces the demand for large training sets.
More broadly, constraining a linear, dimensionality-reduced model to a manifold \(\mathcal {M}\)  has been shown to result in a variety of optimization formulations and techniques which can improve low-rank approximations [1]}.
In figure REF  we compare the performance of standard (REF ) and physics-informed (REF ) DMD
for a simple travelling wave.
We constrain the piDMD model to the shift-invariant symmetry of the underlying system.
As a result, the piDMD model produces an accurate prediction of the future dynamics, whereas the standard DMD model blows up immediately.
This simple example illustrates that embedding basic physics into DMD can substantially improve the algorithm's performance.
